Abstract
The presence of dynamical heterogeneities—that is, nanometrescale regions of molecules rearranging cooperatively at very different rates compared with the bulk^{1,2}—is increasingly being recognized as crucial to our understanding of the glass transition, from the nonexponential relaxation to the divergence of the relaxation times^{3}. Although recent experiments^{4,5,6,7,8,9,10,11} and simulations^{12,13,14} have observed their presence directly, a clear physical picture of their origin is still lacking. Here, we present the first detailed characterization of the statistics of local fluctuations in a simulation of the ageing of a continuousspace, quasirealistic structural glass model. A possible physical mechanism^{15,16,17,18} for the origin of dynamical heterogeneities in the nonequilibrium dynamics of glassy systems predicts universal scaling of the probability distributions of twotime local fluctuations. We find that to a first approximation this scaling is indeed satisfied by our results. We propose to test our results using confocal microscopy and atomic force microscopy experiments.
Similar content being viewed by others
Main
Supercooled liquids approaching the glass transition show increasingly slow dynamics, until eventually they cannot equilibrate in laboratory timescales^{19}. One consequence of this fact is physical ageing, that is, the breakdown of time translation invariance: the correlation C(t,t_{w}) between spontaneous fluctuations of an observable at times t (the final time) and t_{w} (the waiting time) is a nontrivial function of t and t_{w}, as opposed to being a function of the time difference t−t_{w}. In many cases, the twotime correlation C(t,t_{w}) in an ageing system separates into a fast, timetranslationinvariant contribution C_{fast}(t−t_{w}) and a slow contribution C_{slow}(t,t_{w}) (ref. 20): C(t,t_{w})=C_{fast}(t−t_{w})+C_{slow}(t,t_{w}). For some systems, the slow part of the correlation has the form^{20} C_{slow}(t,t_{w})=C_{slow}(h(t)/h(t_{w})), where h(t) is some monotonically increasing function. For example, in the case of domain growth, h(t) is proportional to the domain size^{20}. In what follows, only the slow part of the correlation is considered, and any effects due to the fast part of the dynamics are ignored.
Recently, it has been proven that, in the limit of long times, the dynamics of a class of spinglass models is invariant under global reparametrizations t→h(t) of the time^{15}. This result has been used to predict the existence of a Goldstone mode in the nonequilibrium dynamics, associated with smoothly varying local fluctuations in the reparametrization of the time (refs 16,17). These fluctuations have been physically interpreted to represent local fluctuations of the age of the sample^{16,17}. In the cases where the global twotime correlation shows h(t)/h(t_{w}) scaling, a simple Landautheory approximation for the dynamical action predicts^{16,17,18} that the full probability distribution ρ(C_{r}(t,t_{w})) of local correlations C_{r}(t,t_{w}) depends on the times t,t_{w} only through the values of the global correlation C_{global}(t,t_{w}). However, this scaling of ρ(C_{r}(t,t_{w})) with C_{global}(t,t_{w}) was also found in the coarsening dynamics of the O(N) ferromagnet, where the time reparametrization symmetry is not present^{21}. It is not known whether offlattice, quasirealistic models, describing structural glasses, show the same time reparametrization symmetry as spin glasses, or whether their dynamics shows any evidence of the Goldstone mode associated with this symmetry. A first test for this statement would be to check whether ρ(C_{r}(t,t_{w})) scales with C_{global}(t,t_{w}). If this test fails, then the presence of the time reparametrization symmetry can be excluded.
Simulations of fluctuations in glassforming liquids have mostly focused on the (equilibrium) supercooled liquid, and on determining the spatial correlation of fluctuations between different points in space^{12,13,14}. In ref. 22 the ageing regime was studied, but only the spatial correlations of fluctuations were measured, and in ref. 18 spinglass and kinetically constrained lattice models were studied.
Here, we present the first detailed characterization of the statistics of local fluctuations in the ageing of a continuousspace, quasirealistic structural glass model. Numerically simulating the (nonequilibrium) ageing regime allows us to address many experiments working in this regime that probe dynamical heterogeneities microscopically^{7,8,9,10,11}. We focus here on determining the statistical distribution of fluctuations at one point in space, for various reasons: (1) to make direct contact with experiments using local probes to study dynamical heterogeneities, which also obtain this kind of distribution^{5,6,10}; (2) to obtain additional physical information beyond the second moment of the fluctuations and (3) to test whether the probability distribution of local fluctuations in the ageing regime depends on times only through the value of C_{global}(t,t_{w}).
We probe individual particle displacements along one direction Δx_{j}(t,t_{w})=x_{j}(t)−x_{j}(t_{w}) (where j is the particle index), and also local, coarsegrained twotime functions: the correlator
and the mean square displacement
Here we consider a coarsegraining cubicshaped box B_{r} of side l around the point r in the system, and the sums run over the N(B_{r}) particles present at the waiting time t_{w} in B_{r}. We choose a value of q that corresponds to the main peak in the structure factor S(q) of the system, q=7.2 in LennardJones (LJ) units.
These definitions are inspired by the analogous definitions in the case of spin glasses^{16,17}, and can be applied to analyse data obtained both from simulations and from confocal microscopy experiments. The global quantities C_{global}(t,t_{w}) (incoherent part of the intermediate scattering function) and Δ_{global}(t,t_{w}) (mean square displacement) are defined by extending the sum to the whole system in equations (1) and (2) respectively.
We carried out 250 independent moleculardynamics runs for the binary LJ system of ref. 23, which has a modecoupling critical temperature T_{c}=0.435. A system of 8,000 particles was equilibrated at a temperature T_{0}=5.0, then instantly quenched to T=0.4, and finally it was allowed to evolve for 10^{5} LJ time units. The origin of times was taken at the instant of the quench.
In Fig. 1a,b we present our results for the probability distribution ρ(C_{r}(t,t_{w})) of the local intermediate scattering function for waiting times t_{w}=30.20,…,30,200, and final times t chosen so that C_{global}(t,t_{w})∈{0.1,0.3,0.5,0.7}. We observe that the data approximately collapse for each value of C_{global}(t,t_{w}) (a less clear collapse is observed at constant Δ_{global}(t,t_{w}); details of this comparison will be presented elsewhere). This collapse at constant C_{global}(t,t_{w}) is also observed in simulations in a threedimensional spinglass model, but in the case of the spinglass model the collapse is more precise than here. Unlike the case of the threedimensional spinglass model, the position of the peak in the distribution ρ(C_{r}) is strongly dependent on the value of C_{global}(t,t_{w}). The distribution ρ(C_{r}(t,t_{w})) evolves gradually from being highly skewed and nongaussian for C_{global}(t,t_{w})=0.7 to being unskewed and very close to gaussian for C_{global}(t,t_{w})=0.1. Notice here that the distributions of local observables are also expected to become more gaussian as C_{global}(t,t_{w}) is increased beyond C_{global}(t,t_{w})≈0.7, that is, in the quasiequilibrium regime corresponding to the first step in the twostep relaxation. This is indeed observed in experiments probing fluctuations in dipole moments of nanometrescale regions^{11} and also in our simulations, in the probability distributions ρ(Δx) of onedimensional displacements.
To characterize the weak dependence of the probability distributions on waiting time at fixed C_{global}(t,t_{w}), in Fig. 2a we plot the centred second moment of the distributions ρ(C_{r}) as a function of waiting time, for fixed C_{global}(t,t_{w})∈{0.1,0.3,0.5,0.7}. The dependence on t_{w} is so weak that both a logarithmic form and a powerlaw form (with powers in the range 0.01–0.07) provide a good fit. We can explain the fact that ρ(C_{r}) does show some dependence on t_{w} for fixed C_{global} by the presence of a timedependent dynamic correlation length. As in the case of simulations of spin glasses^{17}, the dynamic correlation length in the present system grows very slowly as a function of t_{w}, but for the timescales of the simulation it is not yet larger than the size of the coarsegraining box used^{24}. Thus, some of the fluctuations are averaged out, and the width of the distribution is reduced. This effect is stronger for shorter t_{w}, consistent with the trend shown in Fig. 2a.
In Fig. 1c,d, we present our results for the probability distribution ρ(Δx(t,t_{w})) of the particle displacements Δx_{j}(t,t_{w})=x_{j}(t)−x_{j}(t_{w}) along one direction. In Fig. 1c, we can observe that these data also approximately collapse for each value of C_{global}(t,t_{w}). In Fig. 1d, we have a closer look at the tails of ρ(Δx(t,t_{w})). We find that the distribution is nongaussian, as was observed in experiments in colloidal glasses in the supercooled regime^{5,6}. We can fit the tails of the distribution with a nonlinear exponential form ρ(Δx)≈Nexp(−Δx/a^{β}), and they become more prominent as t_{w} grows (for constant C_{global}(t,t_{w})). Indeed, as shown in Fig. 2b, the exponent β decreases from β>1 (‘compressed exponential’) at short t_{w} to β≈0.8 (‘stretched exponential’) at much longer t_{w}.
To summarize, we have presented the first detailed characterization of the probability distributions of nonequilibrium fluctuations in the ageing regime in a continuousspace, quasirealistic structural glass model. Our main result is that the probability distributions for the local fluctuating twotime quantities are, to a first approximation, invariant when the global intermediate scattering function C_{global}(t,t_{w}) is kept constant. This behaviour is similar to the behaviour found in the nonequilibrium dynamics of shortrange spinglass models^{16,17} and some kinetically constrained lattice models^{18}, and in the coarsening dynamics of the O(N) model^{21}. As a consequence, our results cannot rule out the presence of a Goldstone mode associated with local fluctuations in the age of the sample, but alternative interpretations are still possible^{21}. Besides this simple scaling, our results provide detailed predictions for the statistical properties of fluctuations in ageing structural glasses. These predictions can be directly tested by applying a similar analysis to experimental data from confocal microscopy in colloidal glass systems^{5,6,7}, and also possibly by analysing atomic force microscopy experiments probing nanoscale polarization fluctuations^{8,9,10,11}.
References
Ediger, M. D. Spatially heterogeneous dynamics in supercooled liquids. Annu. Rev. Phys. Chem. 51, 99–128 (2000).
Sillescu, H. Heterogeneity at the glass transition: A review. J. NonCryst. Solids 243, 81–108 (1999).
Adam, G. & Gibbs, J. H. On the temperature dependence of cooperative relaxation properties in glassforming liquids. J. Chem. Phys. 43, 139–146 (1965).
Kegel, W. K. & Blaaderen, A. V. Direct observation of dynamical heterogeneities in colloidal hardsphere suspensions. Science 287, 290–293 (2000).
Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. Threedimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287, 627–631 (2000).
Weeks, E. R. & Weitz, D. A. Properties of cage rearrangements observed near the colloidal glass transition. Phys. Rev. Lett. 89, 095704 (2002).
Courtland, R. E. & Weeks, E. R. Direct visualization of aging in colloidal glasses. J. Phys. C 15, S359–S365 (2003).
VidalRussell, E., Israeloff, N. E., Walther, L. E. & Alvarez Gomariz, H. Nanometer scale dielectric fluctuations at the glass transition. Phys. Rev. Lett. 81, 1461–1464 (1998).
Walther, L. E., Israeloff, N. E., VidalRussell, E. & Alvarez Gomariz, H. Mesoscopicscale dielectric relaxation at the glass transition. Phys. Rev. B 57, R15112–R15115 (1998).
VidalRussell, E. & Israeloff, N. E. Direct observation of dynamical heterogeneities in colloidal hardsphere suspensions. Nature 408, 695–698 (2000).
Sinnathamby, K. S., Oukris, H. & Israeloff, N. E. Local polarization fluctuations in an aging glass. Phys. Rev. Lett. 95, 067205 (2005).
Glotzer, S. C. Spatially heterogeneous dynamics in liquids: insights from simulation. J. NonCryst. Solids 274, 342–355 (2000).
Kob, W., Donati, C., Plimpton, S. J., Poole, P. H. & Glotzer, S. C. Dynamic heterogeneities in a supercooled LennardJones liquid. Phys. Rev. Lett. 79, 2827–2830 (1997).
Lacevic, N., Starr, F. W., Schroder, T. B. & Glotzer, S. C. Spatially heterogeneous dynamics investigated via a timedependent fourpoint density correlation function. J. Chem. Phys. 119, 7372–7387 (2003).
Chamon, C., Kennett, M. P., Castillo, H. E. & Cugliandolo, L. F. Separation of time scales and reparametrization invariance for aging systems. Phys. Rev. Lett. 89, 217201 (2002).
Castillo, H. E., Chamon, C., Cugliandolo, L. F. & Kennett, M. P. Heterogeneous aging in spin glasses. Phys. Rev. Lett. 88, 237201 (2002).
Castillo, H. E., Chamon, C., Cugliandolo, L. F., Iguain, J. L. & Kennett, M. P. Spatially heterogeneous ages in glassy systems. Phys. Rev. B 68, 134442 (2003).
Chamon, C., Charbonneau, P., Cugliandolo, L. F., Reichman, D. R. & Sellitto, M. Outofequilibrium dynamical fluctuations in glassy systems. J. Chem. Phys. 121, 10120–10137 (2004).
Zallen, R. The Physics of Amorphous Solids (Wiley, New York, 1983).
Bouchaud, J.P., Cugliandolo, L. F., Kurchan, J. & Mézard, M. in Spin Glasses and Random Fields (ed. Young, A. P.) 161–224 (World Scientific, Singapore, 1998).
Chamon, C., Cugliandolo, L. F. & Yoshino, H. Fluctuations in the coarsening dynamics of the O(N) model with N→∞: are they similar to those in glassy systems? J. Stat. Mech. P01006 (2006).
Parisi, G. An increasing correlation length in offequilibrium glasses. J. Phys. Chem. B 103, 4128–4131 (1999).
Kob, W. & Barrat, J.L. Aging effects in a Lennard–Jones glass. Phys. Rev. Lett. 78, 4581–4584 (1997).
Parsaeian, A. & Castillo, H. E. Growth of spatial correlations in the aging of a simple structural glass. Preprint at <http://arxiv.org/abs/condmat/0610789> (2006).
Acknowledgements
H.E.C. especially thanks C. Chamon and L. Cugliandolo for very enlightening discussions over the years and J. P. Bouchaud, S. Glotzer, N. Israeloff, M. Kennett, D. Reichman and E. Weeks for suggestions and discussion. This work was supported in part by the DOE under grant DEFG0206ER46300, by the NSF under grant PHY9907949 and by Ohio University. Numerical simulations were carried out at the Ohio Supercomputing Center and at the Boston University SCV. H.E.C. acknowledges the hospitality of the Aspen Center for Physics.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Castillo, H., Parsaeian, A. Local fluctuations in the ageing of a simple structural glass. Nature Phys 3, 26–28 (2007). https://doi.org/10.1038/nphys482
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys482
This article is cited by

Time reversibility during the ageing of materials
Nature Physics (2024)

When Brownian diffusion is not Gaussian
Nature Materials (2012)

Nanoscale nonequilibrium dynamics and the fluctuation–dissipation relation in an ageing polymer glass
Nature Physics (2010)